Jordan Nested Midpoint Calculus: Exact k-fold Integration Algorithm on Non-associative Algebra Jordan嵌套中道微积分:非结合代数J₃(ℂ)上的k重精确积分算法

Abstract This paper establishes a complete theoretical framework for the nested midpoint calculus on the Exceptional Jordan algebra J(). Key contributions include: (1) Exact k-fold Linear Nesting Formula — strictly proving the exact integration theorem for the product of k-linear factors X₁X₂X, revealing that the bracket structure of the Jordan algebra varies with the depth k (the outermost bracketing differs for k=2 and k≥3); (2) Recursive Analytic Expansion of Jordan Powers — establishing a recursive polynomial expansion algorithm for (a+t·d)ᵐ, applicable to any non-negative integer m; (3) Nonlinear Nesting Generalization — extending the exact algorithm from linear nesting to polynomial nesting of type ZW, covering double and triple cases; (4) Jordan Green’s Theorem — discovering the exact curvature formula for closed path integrals, ∮ Z²dZ = a²b·Δ(U,U,V) − (ab²/2)·{U,V,V}, where Δ is the associator and {·} is the Jordan triple product. All algorithms have been numerically verified to machine precision. 摘要 本文在Exceptional Jordan代数J₃(ℂ)上建立了嵌套中道微积分的完整理论框架。核心贡献包括:(1)k重线性嵌套精确公式——严格证明了k重线性因子乘积X₁X₂Xₙ的精确积分定理,发现Jordan代数的括号结构随层数k变化(k=2与k≥3的最外层括号方式不同);(2)Jordan幂的递归解析展开——建立了(a+t·d)的递归多项式展开算法,适用于任意非负整数m;(3)非线性嵌套推广——将精确算法从线性嵌套推广到ZᵐW型多项式嵌套,覆盖二重和三重情形;(4)Jordan格林定理——发现了闭合路径积分的精确曲率公式Z²dZ = a²b·Δ(U,U,V) − (ab²/2)·{U,V,V},其中Δ为结合子,{·}为Jordan triple product。所有算法均通过数值验证达到机器精度。